(a) What is the probability a randomly selected chip is defective? (b) What is the probability a randomly selected chip was manufactured at Plant B given the chip is defective?
(a) What is the probability a randomly selected chip is defective? (b) What is the probability a randomly selected chip was manufactured at Plant B given the chip is defective?
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Please solve with simple probability rules
![### Probability and Statistics: Manufacturing Defect Rates
#### Problem Statement:
Imtell manufactures computer chips at two facilities in Malaysia. The defective rate for chips manufactured at Plant A is 1.4%. The defective rate for chips manufactured at Plant B is 2.1%. 68% of all chips are manufactured at Plant A.
**Questions:**
(a) What is the probability a randomly selected chip is defective?
(b) What is the probability a randomly selected chip was manufactured at Plant B given the chip is defective?
#### Solution:
To solve these problems, we use the concepts of probability and conditional probability. Let's define the events:
- \( A \): Chip is manufactured at Plant A.
- \( B \): Chip is manufactured at Plant B.
- \( D \): Chip is defective.
Given data:
- \( P(A) = 0.68 \) (probability that a chip is manufactured at Plant A).
- \( P(B) = 0.32 \) (probability that a chip is manufactured at Plant B, since \( P(B) = 1 - P(A) \)).
- \( P(D|A) = 0.014 \) (probability that a chip is defective given it is manufactured at Plant A).
- \( P(D|B) = 0.021 \) (probability that a chip is defective given it is manufactured at Plant B).
**To find:**
(a) The probability that a randomly selected chip is defective, \( P(D) \).
Using the law of total probability:
\[ P(D) = P(D|A)P(A) + P(D|B)P(B) \]
\[ P(D) = 0.014 \times 0.68 + 0.021 \times 0.32 \]
\[ P(D) = 0.00952 + 0.00672 \]
\[ P(D) = 0.01624 \]
So, the probability that a randomly selected chip is defective is 1.624% or 0.01624.
(b) The probability that a randomly selected chip was manufactured at Plant B, given that the chip is defective, \( P(B|D) \).
Using Bayes' theorem:
\[ P(B|D) = \frac{P(D|B)P(B)}{P(D)} \]
\[ P(B|D) = \frac{0.021 \times 0](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F57c1a9e8-096b-4fde-a3f9-c0ea42610dcd%2F008d3f81-4170-4c0b-913e-ff683b2ca468%2Fl7ikf05_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Probability and Statistics: Manufacturing Defect Rates
#### Problem Statement:
Imtell manufactures computer chips at two facilities in Malaysia. The defective rate for chips manufactured at Plant A is 1.4%. The defective rate for chips manufactured at Plant B is 2.1%. 68% of all chips are manufactured at Plant A.
**Questions:**
(a) What is the probability a randomly selected chip is defective?
(b) What is the probability a randomly selected chip was manufactured at Plant B given the chip is defective?
#### Solution:
To solve these problems, we use the concepts of probability and conditional probability. Let's define the events:
- \( A \): Chip is manufactured at Plant A.
- \( B \): Chip is manufactured at Plant B.
- \( D \): Chip is defective.
Given data:
- \( P(A) = 0.68 \) (probability that a chip is manufactured at Plant A).
- \( P(B) = 0.32 \) (probability that a chip is manufactured at Plant B, since \( P(B) = 1 - P(A) \)).
- \( P(D|A) = 0.014 \) (probability that a chip is defective given it is manufactured at Plant A).
- \( P(D|B) = 0.021 \) (probability that a chip is defective given it is manufactured at Plant B).
**To find:**
(a) The probability that a randomly selected chip is defective, \( P(D) \).
Using the law of total probability:
\[ P(D) = P(D|A)P(A) + P(D|B)P(B) \]
\[ P(D) = 0.014 \times 0.68 + 0.021 \times 0.32 \]
\[ P(D) = 0.00952 + 0.00672 \]
\[ P(D) = 0.01624 \]
So, the probability that a randomly selected chip is defective is 1.624% or 0.01624.
(b) The probability that a randomly selected chip was manufactured at Plant B, given that the chip is defective, \( P(B|D) \).
Using Bayes' theorem:
\[ P(B|D) = \frac{P(D|B)P(B)}{P(D)} \]
\[ P(B|D) = \frac{0.021 \times 0
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